The History of the Discovery and Proof
of the Connection between the Spin
of Particles and their Statistics

The spin-statistics theorem concerning the connection the spin of particles and the statistics which they obey is one of the most beautiful results in
mathematical physics. Electrons and other fermion particles are said to have a "spin" of ±½ and bosons like photons have no "spin". More generally
bosons are said to have integral spin and to be the carriers of forces.

Fermions obey the Pauli Exclusion Principle. That means there is at most one particle in a particular state. This accounts for the electron structure of atoms.
There can be any number of bosons in a particular state and they obey Bose-Einstein statistics.

Recently I found that the conventional proof of the spin-statistics theorem depends solely on the mathematical assumptions
involved and not at all on the physics of the situation. The mathematical assumptions are calledl the canonical quantizations of the commutators
and the anti-commutators of the operators of a field. What can be said is that if there is an operator for a field that satisfies the canonical
quantization condition in terms of a commutator then there can be any number of particles of the field in any energy state. On the other hand,
if there is an operator that satisfied the canonical quantization condition in terms of an anti-commutator then at most one particle of the
field can occupy an energy state. The essence of the spin-statistic theorem then involves connecting spin with the canonical quantification
condition for the commutator or the anti-commutator.

In the conventional theory it is alleged that the number operator counts the number of particles in a field. But when the analysis is applied to the
quantum theory of a harmonic oscillator what is found is that the number operator simply gives an integer that is closely related to the principal quantum
number for the oscillator. For the harmonic oscillator the time-independent Schrödinger equation gives a solution that corresponds to a
probability density distribution with (2n+1) maxima lobes.

The energy is quantized as proportional to (n+½), the number of maxima lobe pairs.
The solution can be construed to consist of
n symmetric pairs of lobes and the unpaired lobe at center, but otherwise there are no particles involved. Alternatively one can note that there are
2n minima in the solution, or n minima pairs,

The number operator basically gives the number of energy quanta and if the energy is the sum of particle energies then the number operator
gives the number of particles but otherwise not.

To investigate the topic more thoroughly
I obtained, through interlibrary loan, a book on the spin-statistics theorem. It is by Ian Duck and E.C.G. Shadashan and it is entitled Pauli and the Spin-Statistics Theorem>
It is a very fine work of science history which includes all of the important articles on the spin-statistics theorems in English translation and explications of those
articles. I think it may be the best work of scientific history I have ever seen. It was published in 1997. The copy I have is from the University of San Francisco.
It was checked out only one time in the past 18 years. That was in 2001. Going through it now I find that there were several pages that were not cut apart. This means that no one before
had gone through this book thoroughly. What a testament! What an unutilized source!

The Periodic Table

The story of the Spin-Statistics Theorem should start with the Periodic Table and the attempt of physics to explain its structure.
Here is a standard layout of the periodic table from Wikipedia.

The elements in the same column display the same chemistry. In the first row there are two elements, hydrogen and helium. In the second and third
row there are eight elements. In the fourth and fifth rows there are eighteen elements. The sixth and seventh rows contain 32 elements but with a
different arrangement than in the previous row. There are fifteen elements in the sixth row that all have the same chemistry and that chemistry corresponds
to the third column. They are known as the rare earth elements and as the Lantharide series. The same thing occurs for the seventh row
and the fifteen elements are known as the Actinide series.

So the magic numbers are 2, 8, 18 and 32. Divided by 2 these are 1, 4, 9, and 16, which are usually thought of as simply the squares of the first four integers.
But those numbers arise not as the product of an integer times itself. They arise as the cumulative sums of the first odd numbers

1 = 1
4 = 1 + 3
9 = 1 + 3 +5
16 = 1 + 3 +5 + 7

The odd numbers arise as the number of numbers between −k and +k; i.e., 2k+1, for an integer k.

The atomic number for an element is the number of protons in its nuclei, but it is also the number of its electrons. The electrons are arranged in shells
and its chemistry is determined by the electrons that are outside of its filled shells. For example, sodium has 11 electrons. Two of them fill the first shell.
The next 8 fill the second shell. That leaves only one electron for the third shell. That lonely electron in the third shell is easily lost leaving the sodium atom
to be a sodium ion with a positive charge of 1. On the other hand, fluorine has 9 electrons. Two of them fill the first shell leaving 7 for the second shell. This leaves the second
shell short one of being filled. If the fluorine atom gets that one electron it becomes a ion with a negative charge of 1. When sodium atoms and fluorine atoms
are brought together the single electrons of the sodium atom's outer shells are transferred to the outer shells of the fluorine atoms and the sodium
ions and fluorine ions combine into a crystal of the sodium fluoride compound,

The inert gases of helium, neon, argon. krypton, xenon and radon have fully filled shells and thus experience no ionization and no formation of compounds.

The phenomena of the rare earths arise because a lower energy is involved in putting the additional electrons for Cs 55 and Ba 56 and La 57 into an
outer shell rather than the sixth shell. But the 58th electron through the 71st go into the inner sixth shell. Since the electron arrangements of the outer shells
of the rare earth elements are the same their chemistries are the same. Beyond 71 the electrons go into the outer shell and thus change the chemistry.

The same thing happens for the Actinide series.
.

Pauli's Exclusion Principle

Wolgang Pauli was not the first to articulate the exclusion principle, but he adopted the idea as his own and promoted it and eventually the physics
community identified him with it. It was Edmund C. Stoner who first began to formulate the notion that an electron has a unique identification by
a set of quantum numbers. Stoner and others were working with a set of three quantum numbers. Let n be the pinciple quantum number. There
is a second number l, called the orbital quantum number that can take on values from 0 to (n-1). The third quantum number ml is
called the magnetic quantum number and it can take on values from − l to + l. There are thus (2 l+1) value for
ml. These three numbers would explain the number of states being equal to n². That is when the investigators realized that electron spin with
its two possible values of ±½ was needed to explain the structure of the periodic table. The spin number is designated as ms.

Electron Spin

In 1922 Stern and Gerlach passed a beam of silver ions through a sharply varying magnetic field and
found that the beam separated into two equal beams. This was recognized as something that was
important but puzzling. Three years later Uhlenbeck and Goudsmit at Leiden Unversity published
an article showing that the rotation of the electrical charge of electrons would generate a magnetic
field which would interact with an external magnet field field to cause the deviation of a beam.
The rotation could be in one direction or its opposite so the effect on a beam would be to split it into
two parts. The rate of rotation would be proportional to the angular momentum of the electron
and therefore would be quantized to half of h-bar, ½h.
So the spin is represented as plus or minus one half.

All this could be another example of the false syllogism: A implies B, B is true therefore A is
necessarily true. In this case it is probably true that electrons have rotational spin but that does not necessarily
follow from the Stern-Gerlach experiment. Uhlenbeck and Goudsmit not only derived the Stern-Gerlach result
from electron spin but also the double nature of spectral lines. The energy of an electron would depend upon whether its angular momentum
due to spin is aligned or anti-aligned with the angular momentum due to its rotation about the nucleus. This would account for the slightly
different spectral frequencies. The empirical confirmation of spectroscopy then verifies the actuality of electron spin.

Duck and Sudarshan in their wonderful book "Pauli and the Spin-Statistics Theorem"
notes that a 20 year old student wrote the same material as Uhlenbeck and Goudsmit six months earlier in
1925 but Bohr, Heisenberg and Pauli discouraged him from publishing it. The student,
Ralph Kronig, went on to have a distinguished career in physics and was not resentful of
the Bohr and the Wunderkind.

There was a problem with the idea of electron spin that the speed of the material of electron
at its equator was possibly near or above the speed of light.

Duck and Sudarshan's wonderful book reveals that Uhlenbeck and Goudsmit, due to criticism of their ideas
by a prominent physicist, decided to withdraw their
paper from publication. They asked their adviser Ehrenfest not to send the paper on for publication. He replied,, "I have already sent off
your paper; besides, you are both young enough that you are permitted to make dumbheads of yourselves." The book also reveals that
they got the idea for their paper from a remark by Heisenberg who could not remember where he had heard about it. Heisenberg of course
got the idea from Kronig's paper which he rejected. When Kronig read Uhlenbeck and Goudsmit's paper he said that they had "hidden the
ghosts rather than exorcising them," meaning they had hidden the conceptual problems rather than resolving them.

The Attempts to Prove the Spin-Statistics Theorem

There have been quite a number of purported proofs of
the Spin-Statistics Theorem and the number indicates that
they have not been quite satisfactory. Sometimes the problem
is the lack of mathematical rigor but more often it is a matter
of objections to the assumptions upon which the proof relies.